Vertex Coloring After Parenting in BGE
cont=bge.logic.getCurrentController()
own=cont.owner
mesh=own.meshes[0]
click=cont.sensors["click"]
if click.positive:
    vertex=mesh.getVertex(0,1)
    vertex.color=[1,0,0,1]



This code has no relation to a parent of the (controller owning) object. So this should never be an issue.

I can only guess you execute this code at the armature. An armature has no mesh and will fail when you try to access it (line 3). 

Hint: When using code you should always have a console window open to see any error messages. If you already have ... please post error messages together with your question.